3.25 \(\int \frac{1}{1-\cos ^2(x)} \, dx\)

Optimal. Leaf size=4 \[ -\cot (x) \]

[Out]

-Cot[x]

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Rubi [A]  time = 0.0135559, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3175, 3767, 8} \[ -\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x]^2)^(-1),x]

[Out]

-Cot[x]

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{1-\cos ^2(x)} \, dx &=\int \csc ^2(x) \, dx\\ &=-\operatorname{Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)\\ \end{align*}

Mathematica [A]  time = 0.0022625, size = 4, normalized size = 1. \[ -\cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cos[x]^2)^(-1),x]

[Out]

-Cot[x]

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Maple [A]  time = 0.014, size = 7, normalized size = 1.8 \begin{align*} - \left ( \tan \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)^2),x)

[Out]

-1/tan(x)

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Maxima [A]  time = 0.956744, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^2),x, algorithm="maxima")

[Out]

-1/tan(x)

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Fricas [B]  time = 1.59319, size = 22, normalized size = 5.5 \begin{align*} -\frac{\cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^2),x, algorithm="fricas")

[Out]

-cos(x)/sin(x)

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Sympy [B]  time = 0.596599, size = 14, normalized size = 3.5 \begin{align*} \frac{\tan{\left (\frac{x}{2} \right )}}{2} - \frac{1}{2 \tan{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)**2),x)

[Out]

tan(x/2)/2 - 1/(2*tan(x/2))

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Giac [A]  time = 1.17903, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^2),x, algorithm="giac")

[Out]

-1/tan(x)